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Chapter 7: Problem 64

Explain how to solve a rational equation.

Short Answer

Expert verified
To solve a rational equation, first simplify it if possible, then clear the fractions by multiplying by the Least common denominator. Solve the resulting equation and finally check if the found solutions satisfy the original equation to ensure there are no extraneous solutions.

Step by step solution

01

Simplify the Equation

If the equation can be simplified, do this as a first step. Simplification could be just simplifying a fraction within the equation or the whole equation itself.
02

Clear the fractions

Obtain a simple polynomial equation by multiplying every term by the least common denominator (LCD) of all the fractions.
03

Solve the Resulting Equation

Once fractions are cleared, you will have a polynomial equation. If it is linear, you can solve using regular means. If it is quadratic, use the quadratic formula or factoring methods. Any other higher degree equation can be solved using their relevant methods.
04

Check for Extraneous Solutions

Once the potential solutions have been determined, they need to be checked back in the original equation. Remember that when clearing fractions, solutions can sometimes be introduced that don't actually satisfy the original equation. If a solution does not satisfy the original equation, it is called an extraneous solution.

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Most popular questions from this chapter

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{6}{x-1}-\frac{5}{1-x}$$

Solve: \(x^{2}-12 x+36=0 .\) (Section 6.6, Example 4)

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x-8}{x^{2}-16}-\frac{x-8}{16-x^{2}}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{y-3}{y^{2}-25}+\frac{y-3}{25-y^{2}}$$

add or subtract as indicated. Simplify the result, if possible. $$\frac{3 y^{2}-2}{3 y^{2}+10 y-8}-\frac{y+10}{3 y^{2}+10 y-8}-\frac{y^{2}-6 y}{3 y^{2}+10 y-8}$$
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